For each non-empty wedge at a vertex \(v\)  in the frames and skeleton forests, we insert a new vertex \(v^{\prime }\)  to \(G\) ,
and add the edge between \(v\)  and \(v^{\prime }\) .
Then we remove the edges in the wedge, and reconnect them to \(v^{\prime }\)  instead of \(v\) . Note that, the edges incident to a new vertex \(v^{\prime }\)  are not linkage-edges.
We do this for all non-empty wedges and all vertices in the frames and skeleton forests.
This takes \(O(n)\)  time in total since the total degree of all vertices in the frames and skeleton forests is \(O(n)\) .
Clearly, there is a one-to-one correspondence between two linkages (and weak linkage) on \(G\)  and on \(G_\textsf {mod}\) .
Due to the modification, the running time of [1]} is improved to \(2^{O(k)}\cdot n\cdot \eta \cdot |\Pi |\) .
